Euler's Identity¶

Corollary¶

Setting $z=\pi$ in Euler's formula gives $\mathrm{e}^{\mathrm{i}\pi}+1=0.$

Interpretation¶

Euler's identity is Euler's formula at the angle $\pi$ . For a real angle $\theta$ , $\mathrm{e}^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta$ is the point on the unit circle at angle $\theta$ , and $\theta=\pi$ is half a turn.

The unit circle in the complex plane with an arc of angle pi from the point 1 to the point -1, which is e to the i pi.

The red arc runs anticlockwise from $1=\mathrm{e}^{\mathrm{i}0}$ through $\mathrm{i}=\mathrm{e}^{\mathrm{i}\pi/2}$ and ends at $-1$ : since $\cos\pi=-1$ and $\sin\pi=0$ $0$ , the imaginary part vanishes and $\mathrm{e}^{\mathrm{i}\pi}=-1$ $- 1$ . Adding $1$ to both sides gives $\mathrm{e}^{\mathrm{i}\pi}+1=0$ $0$ .

Read as multiplication, the identity says that multiplying by $\mathrm{e}^{\mathrm{i}\pi}=-1$ is a half-turn about $0$ , that is, negation; two half-turns give $\mathrm{e}^{2\pi\mathrm{i}}=1$ $1$ . The identity is famous for joining the constants $0$ , $1$ , $\pi$ , $\mathrm{e}$ and $\mathrm{i}$ in one short equation, but mathematically it is just a special case of Euler's formula.

Key Results¶

Euler's identity involves three basic arithmetic operations exactly once: addition, multiplication, and exponentiation.
It also links five fundamental mathematical constants: the additive identity $0$ , the multiplicative identity $1$ , the fundamental circle constant $\pi$ , the base of the natural exponential $\mathrm{e}$ , and the imaginary unit $\mathrm{i}$ .

Examples¶

Substitute $\theta=\pi$ into Euler's formula: $\mathrm{e}^{\mathrm{i}\pi}=\cos\pi+\mathrm{i}\sin\pi=-1+\mathrm{i}\cdot0=-1$ $- 1$ . Adding $1$ gives $0$ . Thus multiplication by $\mathrm{e}^{\mathrm{i}\pi}$ sends $z=2+3\mathrm{i}$ to $(-1)(2+3\mathrm{i})=-2-3\mathrm{i}$ $3 i$ , a half-turn about the origin.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Keywords: most beautiful formula Symbols: pi, theta

Last updated: 20 October 2026 03:02